Geometric formulation of scattering amplitudes in N=4 sYM · 2016-12-01 · Geometric formulation...
Transcript of Geometric formulation of scattering amplitudes in N=4 sYM · 2016-12-01 · Geometric formulation...
Geometric formulation of scattering amplitudes in N=4 sYM
Livia Ferro Ludwig-Maximilians-Universität München
„Geometry and Physics“ Workshop in memoriam of Ioannis Bakas
Ringberg Castle, 24.11.2016
Outline Introduction Scattering amplitudes in N=4 sYM Symmetries of amplitudes in planar N=4 Recent formulations for tree-level:
Grassmannian Amplituhedron
Symmetries of the amplituhedron: New differential eqs Volume at tree-level
Open questions
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Scattering amplitudesGauge theories are basis for every model of elementary particles
An =
1
2
n
Scattering amplitudes: central objects in GTs describe interactions between particles
huge number of diagrams high complexity already at two loops symmetries and good formalism can help
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Scattering amplitudesScattering amplitudes: central objects in GTs describe interactions between particlesQCD is the GT of strong interactions
at strong coupling at weak coupling
Can we use another theory?
Problems:
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Scattering amplitudesScattering amplitudes: central objects in GTs describe interactions between particlesQCD is the GT of strong interactions
massivemassless
sYM
N=4 sYM
planar N=4 sYM
(picture of L. Dixon)
Maximally supersymmetric Yang-Mills theory
Different Lagrangians but common properties
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Scattering amplitudes in N=4 sYM
N=4 vs QCD at weak coupling it shares properties of QCD ampls but easier to compute
tree level: gluon ampls are the same loop level: one-loop QCD=sum of susy maximal transcendentality principle
new computational methods can be transferred at high energies QCD -> a conformal limit at strong coupling AdS/CFT can be used
L. Ferro (LMU München) Ringberg Castle, 24.11.2016
Scattering amplitudes in N=4 sYM
N=4 sYM: interacting 4d QFT with highest degree of symmetry
Features: scale invariant hidden symmetries in planar limit - integrable structure triality ampls/Wilson loops/correlation fcs AdS/CFT correspondence
Scattering amplitudes: central objects in GTs describe interactions between particles
L. Ferro (LMU München) Ringberg Castle, 24.11.2016
Scattering amplitudes in N=4 sYMOn-shell supermultiplet described by a superfield:
� = G+ + ⌘A�A +1
2!⌘A⌘BSAB +
1
3!⌘A⌘B⌘C✏ABCD�
D+
1
4!⌘A⌘B⌘C⌘D✏ABCDG�
p2 = 0 () p↵↵̇ = �↵�̃↵̇, q↵A = �↵⌘A
(Tree) Amplitudes labeled by two numbers: number of particles - n helicity - k
MHV tree level[Parke-Taylor]
-
-
++
+
+
+ +
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Scattering amplitudes in N=4 sYM
-> BCFW recursion relations:
On-shell methods
An =X
i,h
Ahi�1
1
P 2i
A�hn�i+1
1̂ n̂
i� 1 i
X P̂iAL AR
„R-invariants“
A6NMHV/A6MHV = [2, 3, 4, 6, 1] + [2, 3, 4, 5, 6] + [2, 4, 5, 6, 1] = [3, 1, 6, 5, 4] + [3, 2, 1, 6, 5] + [3, 2, 1, 5, 4]
ANMHVn = AMHV
n
n�3X
j=2
n�1X
k=j+2
[n, j � 1, j, k � 1, k]
Example:
L. Ferro (LMU München) Ringberg Castle, 24.11.2016
Scattering amplitudes in N=4 sYM
on-shell superspace
twistor superspace momentum-twistor superspace
p
↵↵̇i = x
↵↵̇i � x
↵↵̇i+1
q
↵Ai = ✓
↵Ai � ✓
↵Ai+1
dual superspace
Amplitude Wilson loop<—>duality
WAi = (µ↵
i , �̃↵̇i , ⌘
Ai )
(�↵i , x
↵↵̇i , ✓
↵Ai )
Fourier transform on �↵
i
Later on: bosonized
x1
x2
p1
Incidence relations
(�↵i , �̃
↵̇i , ⌘
Ai )
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Symmetries and ampls in N=4 sYMImportant in discovering the characteristic of amplsSuperconformal symm. Dual superconformal symm.
expected
Yangian symmetry infinite number of „levels“ of generators level-zero generators: level—one generators: + Serre relations
The Yangian Y(g) of the Lie algebra g is generated by j and j(1)
(planar)
* tree-level collinear singularities * broken at loop level
[Witten]
[Drummond,Henn Plefka]
hidden[DHKS]
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-> superconf. gens -> dual superconf. gens
GrassmannianIn momentum twistor space:
Later on… Bosonized momentum twistorsL. Ferro (LMU München) Ringberg Castle, 24.11.2016
c’s: complex parameters forming a kxn matrix (i i+1…i+k-1): determinant of kxk submatrix of c’s GL(k) invariance space of k-planes in n dimensions = Gr(k,n) Yangian invariant
(Arkani-Hamed, Bourjaily, Cachazo, Caron-Huot, Cheung, Goncharov, Hodges, Kaplan,Postnikov,Trnka)(Mason,Skinner)
~~ ~~
~~ ~
AmplituhedronConjecture
the volume of the amplituhedron =
amplitude in planar N=4 sYM
The idea: NMHV tree-level amplitudes = volume of polytope in dual momentum twistor space (Hodges)
different triangulations = different recursion rel.s
(N. Arkani-Hamed, J. Trnka)
(picture of A. Gilmore)
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AmplituhedronThe idea: area of triangle in 2d plane
=1
2
3
Area
I = 1, 2, 3
=
L. Ferro (LMU München) Ringberg Castle, 24.11.2016
AmplituhedronThe idea: area of triangle in 2d plane
=1
2
3
Areaa b
c
:=
Lines in W-space Points in Z-space
„R-invariants“
Amplitude= area of the triangle in dual space
L. Ferro (LMU München) Ringberg Castle, 24.11.2016
AmplituhedronThe idea: area of polytope in 2d plane
1
23
4
L. Ferro (LMU München) Ringberg Castle, 24.11.2016
AmplituhedronThe idea: area of polytope in 2d plane
= -
1
23
43 3
114
2
= [413] - [213]
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AmplituhedronThe idea: area of polytope in 2d plane
= -
1
23
4
1
2 4
32 4
= [412] - [243]= [413] - [213]
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AmplituhedronIn a nutshell
center of mass of 3 points in 2d
~x1
~x2
~x3
·~x
~x =c1~x1 + c2~x2 + c3~x3
c1 + c2 + c3
Interior: ranging over all positive c1, c2, c3
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Amplituhedron
triangle in projective space
Interior: ranging over all positive c1, c2, c3 (GL(1))
·Y
Z1
Z2
Z3
Y I = c1ZI1 + c2Z
I2 + c3Z
I3
Zi =
✓1~xi
◆
I = 1, 2, 3
In a nutshell
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Amplituhedron
„Internal" positivity = interior
Generalizations:triangle -> m-dim.l simplex
In a nutshell
Y I =m+1X
a=1
caZIa
I = 1, 2, ...,m+ 1
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space of k-planes in (k+m) dimensions Y I↵ =
k+mX
a=1
c↵aZIa
I = 1, 2, ..., k +m
->
->
more vertices than (dims+1)
Amplituhedron
Generalizations:most general:
I = 1, 2, ..., k +m
In a nutshell
„Internal" positivity (c) = interior „External" positivity (Z) = convexity
Geometric requirements {
ordered minors > 0
Y I↵ =
nX
a=1
c↵aZIa
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Amplituhedron
Generalizations:
In a nutshell
Positive grassmannian -> tree amplituhedron Ωn,k
Bosonized momentum twistors
Amplituhedron coords :
G(k, k +m)G+(k, n)
G+(k +m,n)
Y I↵ =
nX
a=1
c↵aZIa
L. Ferro (LMU München) Ringberg Castle, 24.11.2016
Amplituhedron
Generalizations:
Positive grassmannian -> tree amplituhedron Ωn,k
G(k, k +m)G+(k, n)
G+(k +m,n)
physics: m=4 tree: k=1 polytope, k>1 more complicated object loops: similar, more complicated formulae
In a nutshell
Y I↵ =
nX
a=1
c↵aZIa
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Amplituhedron
Open questions we want to addresshow to compute volume how to perform the integral symmetries - Yangian?
⌦treen,k
(
Atreen,k (Z) =
Zd4'1...d
4'k
Z�4k(Y, Y0)
Zdk⇥nc
(12...k)(23...k + 1)...(n1...n+ k � 1)
kY
↵=1
Y0 =
0
@O4⇥k
- - -Ik⇥k
1
A
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(LF, T. Lukowski, A. Orta, M. Parisi)
Amplituhedron
Symmetries of Ω ⌦(m)n,k
�k+m(Y↵ �X
a
c↵aZa)
Capelli differential equations(
det
@
@WA⌫aµ
!
1⌫k+11µk+1
⌦(m)n,k (Y, Z) = 0
(LF, T. Lukowski, A. Orta, M. Parisi)
Atreen,k (Z) =
Zdm'1...d
m'k
Z�mk(Y, Y0)
Zdk⇥nc
(12...k)(23...k + 1)...(n1...n+ k � 1)
kY
↵=1
GL(m+k) covariance
GL(1) invariance for Z’s and GL(k) covariance for Y’s
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Amplituhedron
Symmetries of Ω
Start with toy model: m=2
⌦(m)n,k
�k+m(Y↵ �X
a
c↵aZa)
(
(LF, T. Lukowski, A. Orta, M. Parisi)
Atreen,k (Z) =
Zdm'1...d
m'k
Z�mk(Y, Y0)
Zdk⇥nc
(12...k)(23...k + 1)...(n1...n+ k � 1)
kY
↵=1
GL(1) invariance for Z’s and GL(k) covariance for Y’s
GL(m+k) covariance
Capelli differential equations
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A toy model m=2 instead of m=4
k=1
Wi = Zi,W0 = Y
8>><
>>:
@2⌦@WA
i @WBj
= @2⌦@WB
i @WAjP3
A=1 WAj
@⌦@WA
j= ↵j⌦
Pnj=0 W
Aj
@⌦@WB
j= ��AB⌦
homogeneityinvariance
∼Yangian invariance
it satisfies:
What if we start from the differential equations?
⌦(W ) =
Zdnc
c1c2...cn�3(Y � c · Z)
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A toy modelSolution for (m=2, k=1):(following Gelfand, Graev, Retakh)
t2 and t3: dual space
number of particles just in !-fc
-> Generalizes to generic m
shape domain of integration with no singularities t · Y = t · (ciZi) = ci (t · Zi) > 0
⌦ =
Z +1
0dt2dt3
1
(Y 1 + t2Y 2 + t3Y 3)3
nY
i=4
✓(Z1i + t2Z
2i + t3Z
3i )
t2
t3!n
!1
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A toy model
⌦3 =
Z +1
0dt2dt3
1
(Y 1 + t2Y 2 + t3Y 3)3=
1
Y 1Y 2Y 3⌘gf
h123i2
hY 12ihY 23ihY 31i = [123]
t2
t3
[123]
n=3
Solution for (m=2, k=1):
⌦ =
Z +1
0dt2dt3
1
(Y 1 + t2Y 2 + t3Y 3)3
nY
i=4
✓(Z1i + t2Z
2i + t3Z
3i )
L. Ferro (LMU München) Ringberg Castle, 24.11.2016
A toy model
t2
t3
⌦4 =
Z +1
0dt2dt3✓(Z
14 + t2Z
24 + t3Z
34 )
1
(Y 1 + t2Y 2 + t3Y 3)3
�[134]
n=4
Solution for (m=2, k=1):
⌦ =
Z +1
0dt2dt3
1
(Y 1 + t2Y 2 + t3Y 3)3
nY
i=4
✓(Z1i + t2Z
2i + t3Z
3i )
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A toy model
t2
t3
⌦4 = [123] + [134]
t2
t3
[123] - �[134]
n=4
Solution for (m=2, k=1):
⌦ =
Z +1
0dt2dt3
1
(Y 1 + t2Y 2 + t3Y 3)3
nY
i=4
✓(Z1i + t2Z
2i + t3Z
3i )
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Extensions
k=1, generic m
m=4 <-> physics no need to think about triangulation directly in dual space
⌦(m)n,1 =
+1Z
0
m+1Y
A=2
dtA
!m!
(t · Y )m+1
nY
i=m+2
✓ (t · Zi)
Higher helicity integrand not fully fixed can Yangian symmetry help us?
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General ideaIn general:
Capelli differential equations:
det
✓@
@WAi
◆⌦n,k = 0
+ invariance and scaling
Question: find a function satisfying the requirements Grassmannian integral satisfies the eqs for any k solve the diff. eqs directly in dual space no need to think about triangulation
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Conclusions
Amplituhedron is conjectured to give a geometric interpretation of the amplitudes
for planar N=4 sYM
A lot of work still has to be done!
how to evaluate volume for k>1 and for loops? is Yangian symmetry preserved? do Capelli diff. eqs correspond to a symmetry? new recursion relations?
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